A NEW PROOF OF SZAB´ O'S THEOREM ON THE RIEMANN-METRIZABILITY OF BERWALD MANIFOLDS

The starting point of the famous structure theorems on Berwald spaces due to Z.I. Szabo (4) is an observation on the Riemann-metrizability of positive definite Berwald manifolds. It states that there always exists a Riemannian metric on the underlying manifold such that its Levi-Civita con- nection is just the canonical connection of the Berwald manifold. In this paper we present a new elementary proof of this theorem. After constructing a Rie- mannian metric by the help of integration of the canonical Riemann-Finsler metric on the indicatrix hypersurface it is proved that in case of Berwald man- ifolds the canonical connection and the Levi-Civita connection coincide.